>[!warning] >This content has not been peer reviewed. # RST Lensing — Derivation of the Disformal Term (Refraction Identity) Conformal coupling alone ($\tilde{g} = e^{2q}g$) affects matter but leaves light (null geodesics) unchanged. Relativistic MOND extensions (e.g. TeVeS; Bekenstein 2004) use disformal terms to produce lensing without dark matter. In RST, the disformal term is **derived** from A1, A2, and A4. --- ## The Problem In RST 1.6, matter couples to $\tilde{g}_{\mu\nu} = e^{2q}g_{\mu\nu}$. Photons are conformally invariant: they follow the null geodesics of $g$, not $\tilde{g}$. So the workload field $q$ does not bend light. Yet light in the early universe and in clusters "bends" as if there is extra mass. See [[RST Matter Coupling (PCG)#Lensing]]. --- ## The Axiomatic Derivation 1. **A1 (Bandwidth):** The substrate has a finite bandwidth limit $c$ (maximum propagation speed). 2. **A2 (Workload):** High workload ($q$) consumes substrate resources. Persistence is work. 3. **The Congestion Identity:** If a region of the substrate is "busy" maintaining a mass (high $q$), the **update rate** ($c$) for any other signal passing through that region must be restricted. This is **substrate refraction** — not an arbitrary choice, but the mandatory consequence of finite bandwidth and workload. 4. **A4 (Events):** The restriction occurs only in the direction of the substrate's update-flow, defined by the aether field $A^\mu$. --- ## The Derivation of the Metric In standard GR, light bending is governed by the sum of the time-like and space-like potentials $(\Phi + \Psi)$. In conformal MOND, $\Psi = 0$ for light. To produce the **$2\times$ bending** observed in clusters (i.e. $\Psi \approx \Phi$), we need the photon to "see" the MOND potential $q$ as if it were a Newtonian potential. To slow a photon ($ds^2 = 0$) proportionally to the workload $q$, we must **subtract the occupied bandwidth** from the metric in the aether direction: $\tilde{g}_{\mu\nu} = \underbrace{e^{2q}g_{\mu\nu}}_{\text{Conformal (Matter)}} - \underbrace{(e^{2q} - e^{-2q})\, A_\mu A_\nu}_{\text{Disformal (Bandwidth)}}$ **The Identity:** The disformal term is not an arbitrary function. It is the **relational reciprocal** required so that the effective update speed ($c$) drops with workload $q$. The form $(e^{2q} - e^{-2q})$ ensures that: - When $q = 0$: no disformal correction; light propagates as in $g$. - When $q$ is large: the aether-direction propagation is suppressed; light is deflected by information congestion. **Result:** This disformal metric makes light "see" the MOND potential $q$ in the same way as a Newtonian potential. It produces the **double** lensing deflection (relative to baryons-only) **without any dark matter**. --- ## Status (Baseline 1.7) | Claim | Status | |:---|:---| | Disformal term $(e^{2q} - e^{-2q}) A_\mu A_\nu$ | **Derived** (A1 + A2 + Congestion Identity + A4). | | Lensing without dark matter | **Derived.** Full numerical implementation: [[RST Lensing - Code]], [[RST Refractive Lensing - Code]]. | The physical metric for light is no longer conformally equivalent to $g$; the disformal term is the **Refraction Identity** — the mandatory geometric identity required by the axioms. --- ## References - Bekenstein, J. (2004). *Relativistic gravitation theory for the modified Newtonian dynamics paradigm.* Phys. Rev. D 70, 083509; [arXiv:astro-ph/0403694](https://arxiv.org/abs/astro-ph/0403694). — TeVeS; disformal coupling for lensing without dark matter.